We report a new analytical method for finding the exact solution of homogeneous linear ordinary differential equations with arbitrary order and variable coefficients. The method is based on the definition of jump transfer matrices and their extension into limiting differential form. The approach reduces the nth-order differential equation to a system of n linear differential equations with unit...

This paper presents a numerical matrix method based on Bernstein polynomials (BPs) for approximate the solution of a system of m-th order nonlinear Volterra integro-differential equations under initial conditions. The approach is based on operational matrices of BPs. Using the collocation points,this approach reduces the systems of Volterra integro-differential equations associated with the giv...

We formulate a series of strong ellipticity inequalities for equilibrium equations the gradient elasticity up to Nth order. Within this model continuum, there exists deformation energy introduced as an objective function gradients As result, constitute system 2N-order nonlinear partial differential (PDEs). Using these boundary-value problem with Dirichlet boundary conditions, we prove positive ...